scholarly journals THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

2010 ◽  
Vol 21 (11) ◽  
pp. 1387-1399
Author(s):  
NING ZHANG
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Loop Space ◽  
Riemann Sphere ◽  
Loop Group ◽  
Picard Group ◽  
Line Bundles ◽  
Projective Embedding ◽  
Infinite Dimensional ◽  
Holomorphic Line Bundles

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.

scholarly journals Differentiability of the evolution map and Mackey continuity

2019 ◽  
Vol 31 (5) ◽  
pp. 1139-1177
Author(s):  
Maximilian Hanusch
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Continuity Property ◽  
Fréchet Space ◽  
Frechet Space ◽  
Regularity Problem ◽  
Infinite Dimensional ◽  
Made In

AbstractWe solve the differentiability problem for the evolution map in Milnor’s infinite-dimensional setting. We first show that the evolution map of each {C^{k}}-semiregular Lie group G (for {k\in\mathbb{N}\sqcup\{\mathrm{lip},\infty\}}) admits a particular kind of sequentially continuity – called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially and the Mackey continuous context. We furthermore conclude that if the Lie algebra of G is a Fréchet space, then G is {C^{k}}-semiregular (for {k\in\mathbb{N}\sqcup\{\infty\}}) if and only if G is {C^{k}}-regular.

scholarly journals Irreducible Unitary Representations Concerning Homogeneous Holomorphic Line Bundles Over Elliptic Orbits

2018 ◽  
Vol 10 (4) ◽  
pp. 62
Author(s):  
Nobutaka Boumuki ◽  
Tomonori Noda
Keyword(s):  
Line Bundle ◽  
Cross Sections ◽  
Lie Group ◽  
Vector Subspace ◽  
Line Bundles ◽  
Semisimple Lie Group ◽  
Complex Vector ◽  
Elliptic Orbits ◽  
Irreducible Unitary Representations ◽  
Holomorphic Line Bundles

In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

Infinite Dimensional DeWitt Supergroups and their Bodies

2014 ◽  
Vol 57 (2) ◽  
pp. 283-288 ◽  
Author(s):  
Ronald Fulp
Keyword(s):  
Banach Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Grassmann Algebra ◽  
The Body ◽  
Exponential Mapping ◽  
Infinite Dimensional ◽  
Group Operation ◽  
Banach Lie Group ◽  
Quotient Manifold

AbstractFor Dewitt super groups G modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group BG compatible with the group operation on G, then, generically, the kernel K of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra κ has the property that for each a ∊ κ ada has a zero spectrum. We also show that the exponential mapping from κ to K is surjective and that K is a quotient manifold of the Banach space κ via a lattice in κ.

scholarly journals Projective Bundles on Infinite-Dimensional Complex Spaces

2004 ◽  
Vol 11 (1) ◽  
pp. 43-48
Author(s):  
E. Ballico
Keyword(s):  
Banach Space ◽  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Infinite Dimensional ◽  
Complex Spaces ◽  
Projective Bundles ◽  
Holomorphic Embeddings ◽  
Holomorphic Line Bundles

Abstract Let V be a complex localizing Banach space with countable unconditional basis and E a rank r holomorphic vector bundle on P(V). Here we study the holomorphic embeddings of P(E) into products of projective spaces and the holomorphic line bundles on P(E). In particular we prove that if r ≥ 3, then H 1(P(E), L) = 0 for every holomorphic line bundle L on P(E).

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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